Not every line goes through the origin, but every line has a steepness. Building on unit rate as slope, slope measures how fast one quantity changes compared with another.
Por qué funciona
When something changes at a constant rate, equal steps in $x$ always bring equal changes in $y$. So the ratio $\dfrac{\text{change in } y}{\text{change in } x}$ is the same no matter which two points you pick. That one number describes the whole line, and it is the rate of change.
Idea clave
$$\text{slope} = \frac{\text{change in } y}{\text{change in } x} = \text{rate of change}$$ It tells how much $y$ changes for each 1 that $x$ increases. Positive: $y$ goes up. Negative: $y$ goes down. Zero: $y$ stays the same.
Worked examples
Ejemplo resuelto: From a table
A candle burns down: $$\begin{array}{c|cccc} \text{hours} & 0 & 1 & 2 & 3 \\ \hline \text{height (cm)} & 20 & 18 & 16 & 14 \end{array}$$ What is the rate of change?
- Each hour, the height changes by $18 - 20 = -2$ cm.
- It is the same for every pair of neighboring columns.
Respuesta: $-2$ cm per hour
Ejemplo resuelto: Steps bigger than 1
$$\begin{array}{c|cccc} \text{minutes} & 0 & 5 & 10 & 15 \\ \hline \text{liters} & 3 & 23 & 43 & 63 \end{array}$$ What is the rate of change?
- Change in $y$: $23 - 3 = 20$. Change in $x$: $5 - 0 = 5$.
- Rate: $\frac{20}{5} = 4$.
Respuesta: 4 liters per minute
Ejemplo resuelto: Two data points
A car has gone 150 miles after 2 hours and 330 miles after 5 hours, at a steady speed. What is the rate of change?
- Change in distance: $330 - 150 = 180$ miles.
- Change in time: $5 - 2 = 3$ hours.
- $\frac{180}{3} = 60$.
Respuesta: 60 miles per hour
Error común
Thinking slope is a height. A slope of 3 does not mean the line is 3 units high; it means the line rises 3 for every 1 to the right. A line can sit high on the graph and still have slope 0, if it is flat.